Solving One-Variable Equations by Collecting Terms

Solving one-variable equations by collecting terms is a technique that helps simplify and solve equations efficiently. The process involves combining like terms, moving constants from one side of the equation to the other, and isolating the variable. By applying this method, you can solve equations that may appear complicated at first glance, turning them into simpler ones.


What Are Like Terms?

Before we dive into the method of collecting terms, it’s essential to understand what like terms are. Like terms are terms with the same variable raised to the same power. For example:

  • 3x3x and 5x5x are like terms because they contain the variable x raised to the first power.
  • 4y24y^2 and −7y2-7y^2 are like terms because they involve the variable y squared.

On the other hand, 3x3x and 4y4y are not like terms because they involve different variables.


Step-by-Step Process for Solving One-Variable Equations by Collecting Terms

 Simplify Each Side of the Equation

To begin solving an equation by collecting terms, simplify both sides by combining like terms. For example, consider the equation:

3x+5−2x=93x + 5 - 2x = 9Here, the terms 3x and -2x are like terms because both involve the variable x. Combine these terms:

(3x−2x)+5=9(3x - 2x) + 5 = 9This simplifies to:

x+5=9x + 5 = 9Step 2: Move Constants to One Side

Once you have simplified the terms on each side, isolate the variable by moving constants to the other side of the equation. To do this, subtract 5 from both sides of the equation:

x=9−5x = 9 - 5x=4x = 4Thus, the solution to the equation is x = 4.

 

Check the Solution

After solving the equation, always substitute the value of x back into the original equation to verify your solution. For the equation 3x+5−2x=93x + 5 - 2x = 9, substitute x = 4:

3(4)+5−2(4)=93(4) + 5 - 2(4) = 912+5−8=912 + 5 - 8 = 99=99 = 9Since both sides are equal, the solution x = 4 is correct.

Example 2: Solving Equations with More Complex Terms

Let’s now consider a more complex equation that requires collecting terms on both sides. Solve the equation:

4x+3−2x=2x+74x + 3 - 2x = 2x + 7First, combine the like terms on both sides. On the left side, combine the terms with x:

(4x−2x)+3=2x+7(4x - 2x) + 3 = 2x + 7This simplifies to:

2x+3=2x+72x + 3 = 2x + 7Next, subtract 2x from both sides to eliminate the variable term from one side:

3=73 = 7Here, you’re left with a contradiction (3 ≠ 7), meaning the original equation has no solution.


 Deal with Special Cases

In some cases, collecting terms will lead to special situations:

  • If, after simplifying, you get an equation where both sides are equal (e.g., 2x+3=2x+32x + 3 = 2x + 3), this implies that the equation has infinitely many solutions. This occurs because the equation is true for all values of x.
  • The equation has no solution if you get a contradiction, as in the previous example.


Solving one-variable equations by collecting terms is a crucial skill in algebra that helps you simplify and quickly solve equations. The process involves combining like terms, moving constants, and isolating the variable. By following these steps, you can handle both simple and complex equations. Always check your solution to ensure its correctness, and remember that sometimes equations may have no solution or infinitely many solutions.



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